This activity involves rounding four-digit numbers to the nearest thousand.

Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?

Are these statements relating to calculation and properties of shapes always true, sometimes true or never true?

Are these statements always true, sometimes true or never true?

Here are two kinds of spirals for you to explore. What do you notice?

Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?

How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

In this problem we are looking at sets of parallel sticks that cross each other. What is the least number of crossings you can make? And the greatest?

Try adding together the dates of all the days in one week. Now multiply the first date by 7 and add 21. Can you explain what happens?

An investigation that gives you the opportunity to make and justify predictions.

Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?

What happens when you round these three-digit numbers to the nearest 100?

Put the numbers 1, 2, 3, 4, 5, 6 into the squares so that the numbers on each circle add up to the same amount. Can you find the rule for giving another set of six numbers?

Are these statements relating to odd and even numbers always true, sometimes true or never true?

Watch this video to see how to roll the dice. Now it's your turn! What do you notice about the dice numbers you have recorded?

This challenge encourages you to explore dividing a three-digit number by a single-digit number.

Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.

This problem challenges you to find out how many odd numbers there are between pairs of numbers. Can you find a pair of numbers that has four odds between them?

Strike it Out game for an adult and child. Can you stop your partner from being able to go?

Nim-7 game for an adult and child. Who will be the one to take the last counter?

Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.

Find a route from the outside to the inside of this square, stepping on as many tiles as possible.

Can you find all the ways to get 15 at the top of this triangle of numbers?

This task follows on from Build it Up and takes the ideas into three dimensions!

Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.

What happens when you round these numbers to the nearest whole number?

Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?

In this calculation, the box represents a missing digit. What could the digit be? What would the solution be in each case?

This challenge focuses on finding the sum and difference of pairs of two-digit numbers.

In this game for two players, the idea is to take it in turns to choose 1, 3, 5 or 7. The winner is the first to make the total 37.

This challenge is about finding the difference between numbers which have the same tens digit.

Got It game for an adult and child. How can you play so that you know you will always win?

Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.

Two children made up a game as they walked along the garden paths. Can you find out their scores? Can you find some paths of your own?

Here are some arrangements of circles. How many circles would I need to make the next size up for each? Can you create your own arrangement and investigate the number of circles it needs?

The NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails.

For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?

While we were sorting some papers we found 3 strange sheets which seemed to come from small books but there were page numbers at the foot of each page. Did the pages come from the same book?

Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.

One block is needed to make an up-and-down staircase, with one step up and one step down. How many blocks would be needed to build an up-and-down staircase with 5 steps up and 5 steps down?

This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning.

Can you dissect an equilateral triangle into 6 smaller ones? What number of smaller equilateral triangles is it NOT possible to dissect a larger equilateral triangle into?

Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.